Econ 805 Problem Set 3 Solution
1. (Gibbons 1.2). In the following game, what strategies survive IESDS (iterated
elimination of strictly dominated strategies)? What are the pure strategy Nash equilibria?
LCR
T
2
,
01
,
14
,
2
M
3
,
4
1
,
22
,
3
B
1
,
3
0
,
23
,
0
Answer: for player 1,
B
is strictly dominated by
T
. Sowecande
letetherowo
f
B
.
Thenforp
layer2
,
C
is strictly dominated by
R
. No further strategies can be deleted.
Therefore, the strategies that survive IESDS are:
(
T,M
)
and
(
L, R
)
.
ThepurestrategyNEo
fth
isgameare
(
M,L
)
and
(
T,R
)
(see the underscore in the
bi-matrix).
2. Mascolell 8.B.5.
Answer: (i) First, any quantity
q>q
m
=
a
−
c
2
b
(the monopoly quantity) is strictly
dominated by
q
m
. Then the remaining strategy set for each
f
rm is
[0
,q
m
]
.N
o
wy
o
u
can calculate the best response to
q
m
.Y
o
uw
i
l
lg
e
t
R
(
q
m
)=
a
−
c
2
b
−
q
m
2
.. One can show
that any
q<b
(
q
m
)
is strictly dominated by
R
(
q
m
)
. Now the relevant strategy set is
[
R
(
q
m
)
,q
m
]
. Now you can calculate the best response to
R
(
q
m
)
. You can easily get that
R
2
(
q
m
)=
a
−
c
2
b
−
R
(
q
m
)
2
. Again you can prove that any quantity
q>R
2
(
q
m
)
is strictly
dominated by
R
2
(
q
m
)
. Now the relevant strategy set shrinks to
[
R
(
q
m
)
,R
2
(
q
m
)]
.I
fy
o
u
continue this process, after
2
n
rounds of IESDS, the remaining strategy set is
[
R
2
n
−
1
(
q
m
)
,
R
2
n
(
q
m
)]
. Nowyouon
lyneedtoshowthat
R
n
(
q
m
)
converges as
n
goes to in
f
nity. You
can show that
R
n
(
q
m
)=
2
n
−
2
n
−
1
+2
n
−
2
+
...
+(
−
1)
n
2
n
−
n
2
n
q
m
=[
1
−
1
2
+
1
4
−
...